MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfinufil Structured version   Visualization version   GIF version

Theorem cfinufil 23423
Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 23388 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
cfinufil (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elpwi 4608 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 ufilb 23401 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
32adantr 481 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
4 ufilfil 23399 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
54adantr 481 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹 ∈ (Fil‘𝑋))
6 filfinnfr 23372 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋𝑥) ∈ 𝐹 ∧ (𝑋𝑥) ∈ Fin) → 𝐹 ≠ ∅)
763exp 1119 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐹 → ((𝑋𝑥) ∈ Fin → 𝐹 ≠ ∅)))
87com23 86 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
95, 8syl 17 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
109imp 407 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅))
113, 10sylbid 239 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 𝐹 ≠ ∅))
1211necon4bd 2960 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ( 𝐹 = ∅ → 𝑥𝐹))
1312ex 413 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ( 𝐹 = ∅ → 𝑥𝐹)))
1413com23 86 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
151, 14sylan2 593 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
1615ralrimdva 3154 . . 3 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
174adantr 481 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18 uffixsn 23420 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ∈ 𝐹)
19 filelss 23347 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
2017, 18, 19syl2anc 584 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ⊆ 𝑋)
21 dfss4 4257 . . . . . . . . . . 11 ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
2220, 21sylib 217 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23 snfi 9040 . . . . . . . . . 10 {𝑦} ∈ Fin
2422, 23eqeltrdi 2841 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25 difss 4130 . . . . . . . . . . 11 (𝑋 ∖ {𝑦}) ⊆ 𝑋
26 filtop 23350 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
27 elpw2g 5343 . . . . . . . . . . . 12 (𝑋𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2817, 26, 273syl 18 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2925, 28mpbiri 257 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30 difeq2 4115 . . . . . . . . . . . . 13 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
3130eleq1d 2818 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32 eleq1 2821 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3331, 32imbi12d 344 . . . . . . . . . . 11 (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋𝑥) ∈ Fin → 𝑥𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3433rspcv 3608 . . . . . . . . . 10 ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3529, 34syl 17 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3624, 35mpid 44 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37 ufilb 23401 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3820, 37syldan 591 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3918pm2.24d 151 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 𝐹))
4038, 39sylbird 259 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 𝐹))
4136, 40syld 47 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ¬ 𝑦 𝐹))
4241impancom 452 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → (𝑦 𝐹 → ¬ 𝑦 𝐹))
4342pm2.01d 189 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → ¬ 𝑦 𝐹)
4443eq0rdv 4403 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → 𝐹 = ∅)
4544ex 413 . . 3 (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → 𝐹 = ∅))
4616, 45impbid 211 . 2 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
47 rabss 4068 . 2 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹))
4846, 47bitr4di 288 1 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  {crab 3432  cdif 3944  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627   cint 4949  cfv 6540  Fincfn 8935  Filcfil 23340  UFilcufil 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1o 8462  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fbas 20933  df-fg 20934  df-fil 23341  df-ufil 23396
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator